The Once But Not Twice Differentiability of the Policy Function

Abstract

THE GOAL OF THIS PAPER is to study the differentiability of the policy function or, in other words, the twice differentiability of the value function, in onedimensional dynamic programming problems. Here we treat the case commonly used in Economics where the objective function is a concave function. The need for the twice differentiability of the value function has been noticed a long time ago. Pontryagin et al. (1962, p. 73, last paragraph) shows it to be necessary to derive the maximum principle from the Bellman equation, and gives a counterexample for nonconcave problems. The twice differentiability of the value function is also important if one wants to do comparative statics or if one wants to have smooth coefficients in Bellman's partial differential equation. Another important application has been given recently by Kehoe, Levine, and Romer (1989) to obtain the finiteness of the number of equilibria in an economy with an infinite number of goods and a finite number of consumers. For other motivations as well as background for this paper, see Stokey and Lucas (1989). More formally, in the second section of this paper we introduce the basic facts and notations to be used latter. The third section is devoted to showing that the policy function is C1 and therefore that the value function is C2, if the policy function is increasing. For a multidimensional generalization the reader is referred to Santos (1991). In Section 4 we show, by means of a counterexample, that the policy function might not be a twice differentiable function and hence that the value function might not be three times differentiable, even if the objective is three times continuously differentiable and strongly concave and the